Emergent gravity and higher spin on covariant quantum spaces

Transcription

1 Emergent gravity and higher spin on covariant quantum spaces Harold Steinacker Department of Physics, University of Vienna Corfu, september 2017

2 Motivation requirements for fundamental theory simple, constructive finite dof (per volume ), pre-geometric gauge theory string theory: far-reaching, but issues: compactification definition (why? landscape? testable?)

3 Motivation requirements for fundamental theory simple, constructive finite dof (per volume ), pre-geometric gauge theory string theory: far-reaching, but issues: compactification definition (why? landscape? testable?) Matrix Models as fundamental theories of space-time & matter string theory (IKKT model (this talk!), BFSS model) NC gauge theory

4 outline: quantum (NC) spaces from matrix models 4D covariant quantum spaces: fuzzy S 4 fluctuations higher spin theory in M.M. metric, vielbein; towards gravity cosmological space-times HS, arxiv: M. Sperling, HS arxiv: M. Sperling, HS arxiv: HS, arxiv: xxx

5 The IKKT model IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996 ( ) S[X, Ψ] = Tr [X a, X b ][X a, X b ]η aa η bb + Ψγ a [X a, Ψ] X a = X a Mat(N, C), a = 0,..., 9, N large gauge symmetry X a UX a U 1, SO(9, 1), SUSY proposed as non-perturbative definition of IIB string theory origins: quantized Schild action for IIB superstring reduction of 10D SYM to point, N large N = 4 SYM on noncommutative R 4 θ

6 leads to matrix geometry : ( NC geometry) S E Tr[X a, X b ] 2 config s with small [X a, X b ] 0 dominate i.e. almost-commutative configurations, geometry quasi-coherent states x, minimize a x X a 2 x X a diag., spectrum =: M R 10 x X a x δ(x x )x a, x M NC branes embedded in target space R 10 X a x a : M R 10

7 how to preserve Lorentz / SO(4) covariance in 4D? obstacle: NC spaces: 0 [X µ, X ν ] =: iθ µν θ µν breaks Lorentz invariance (in D > 2) fully covariant fuzzy four-sphere S 4 N Grosse-Klimcik-Presnajder 1996; Castelino-Lee-Taylor; Ramgoolam; Kimura; Hasebe; Medina-O Connor; Karabail-Nair; Zhang-Hu 2001 (QHE!)... price to pay: internal structure higher spin theory here: work out higher spin modes on S 4 N higher-spin gauge theory from matrix models

8 covariant fuzzy four-spheres 5 hermitian matrices X a, a = 1,..., 5 acting on H N Xa 2 = R 2 a covariance: X a End(H N ) transform as vectors of SO(5) [M ab, X c ] = i(δ ac X b δ bc X a ), [M ab, M cd ] = i(δ ac M bd δ ad M bc δ bc M ad + δ bd M ac ). denote M ab... so(5) generators acting on H N [X a, X b ] =: iθ ab

9 covariant fuzzy four-spheres 5 hermitian matrices X a, a = 1,..., 5 acting on H N Xa 2 = R 2 a covariance: X a End(H N ) transform as vectors of SO(5) [M ab, X c ] = i(δ ac X b δ bc X a ), [M ab, M cd ] = i(δ ac M bd δ ad M bc δ bc M ad + δ bd M ac ). M ab... so(5) generators acting on H N denote [X a, X b ] =: iθ ab oscillator construction: Grosse-Klimcik-Presnajder 1996;... X a = ψ α(γ a ) α βψ β, [ψ β, ψ α] = δα β acting on H N = ψ α 1...ψ α N 0 = (C 4 ) SN = (0, N)so(5)

10 relations: Θ ab = r 2 M ab X a X a = R r 2 N 2 ɛ abcde X a X b X c X d X e = (N + 2)R 2 r 3 (volume quantiz.) geometry from coherent states p : {p a = p X a p } = S 4 closer inspection: degeneracy of coherent states at internal fuzzy S 2 N+1 fiber

11 semi-classical picture: hidden bundle structure CP 3 ψ S 4 x a = ψ + Γ a ψ fuzzy case: [Ψ, Ψ ] = δ Ho-Ramgoolam, Medina-O Connor, Abe,... X a = Ψ Γ a Ψ M ab = Ψ Σ ab Ψ...functions on fuzzy CP 3 N fuzzy SN 4 is really fuzzy CP3 N, hidden extra dimensions S2! Poisson tensor θ µν (x, ξ) i[x µ, X ν ] rotates along fiber ξ S 2! is averaged [θ µν (x, ξ)] 0 = 0 over fiber local SO(4) preserved, 4D covariant quantum space

12 fields and harmonics on S 4 N functions on S 4 N : φ End(H N ) = s n N (n, 0) modes = scalar functions on S 4 : (n s, 2s) = φ(x) = φ a1...a n X a 1...X an = (n, 2) modes = selfdual 2-forms on S 4 etc. φ bc (X)M bc = φ a1...a nb;cx a 1...X an M bc = tower of higher spin modes, s = 0, 1, 2,..., N from twisted would-be KK modes on S 2 (local SO(4) acts non-trivially on S 2 fiber)

13 relation with spin s fields: isomorphism (, 2s) = T S s S 4 φ (s) = φ (s) b 1...b s;c 1...c s (x) θ b 1c 1... θ bscs φ (s) c 1...c s (x) = φ (s) b 1...b s;c 1...c s x b 1... x bs... symbol of φ C s M. Sperling & HS, arxiv: (, 2s) = symm., traceless, tang., div.-free rank s tensor field on S 4 φ c1...c s (x)x c i = 0, φ c1...c s (x)g c 1c 2 = 0, c i φc1...c s (x) = 0. functions on S 4 N = hs - valued functions on S 4 cf. Vasiliev theory!

14 local description: pick north pole p S 4 tangential & radial generators ( ) X a X µ =, µ = 1,..., 4...tangential coords at p X 5 separate SO(5) into SO(4) & translations ( ) M ab M µν P = P µ where P µ = M µ5 0 rescale P µ = 1 R g µνp ν (cf. Wigner contraction) Poisson algebra {P µ, X ν } = δµ, ν {P µ, P ν } = 1 M µν R 2 0 {X µ, X ν } =: θ µν = ir 2 M µν 0 (cf. Snyder space)

15 local form of s = 2 modes: φ (2) = φ a1...a nbc;dex a 1...x an θ bd θ ce =: h µν (x)p µ P ν + ω µ:αβ (x)p µ M αβ + Ω αβ;µν (x)m αβ M µν where ω µ;αβ = n+1 (n+2)(n+3) ( αh µβ β h µα ) Ω αβ;µν = 1 (n+2)(n+3) R αβµν[h]... (lin.) spin connection and curvature determined by h µν (irrep of so(5) ) action for higher spin gauge theory on SN 4? matrix models!

16 IKKT model S = Tr( [Y a, Y b ][Y a, Y b ] + µ 2 Y a Y a ) eom: ( + µ2 2 )X a = 0, = [X a, [X a,.]] fact: S 4 N is solution (for µ2 < 0) (cf. HS arxiv: ) add fluctuations Y a = X a + A a expand action to second oder in A a S[Y ] = S[X] + 2 g 2 TrA a ( ) ( + µ 2 )δb a + 2[[X a, X b ],. ] [X a, [X b,.]] A b fluctuations A describe gauge theory (NCFT) on M ( open strings ending on M) effective metric G µν (x) θ µµ θ νν g µ ν (review: H.S. arxiv: ) matrix model provides off-shell formulation of hs gauge theory

17 tangential fluctuations at p S 4 : A µ = θ µν A ν where A ν (x) = A ν (x) +A νρ (x)p ρ + A νρσ (x)m ρσ +... }{{} A νab M ab...so(5) connection... hs - valued gauge field rank 2 tensor field A νρ (x) = 1 2 (h νρ + a νρ ) h νρ = h ρν... metric fluctuation rank 3 tensor field A νρσ (x)m ρσ... so(4) connection rank 1 field A ν (x)... U(1) gauge field

18 gauge transformations: Y a UY a U 1 = U(X a + A a )U 1 leads to (U = e iλ ) δa a = i[λ, X a ] + i[λ, A a ] expand Λ = Λ Λ abm ab U(1) SO(5)... - valued gauge trafos diffeos from δ v := i[v ρ P ρ,.] δh µν = ( µ v ν + ν v µ ) v ρ ρ h µν + (Λ h) µν δa µρσ = 1 2 µλ σρ (x) v ρ ρ A µρσ + (Λ A) µρσ etc.

19 metric and vielbein consider scalar field φ = φ(x) kinetic term vielbein (= transversal fluctuation A a (X)) [X α, φ][x α, φ] e α φe α φ = γ µν µ φ ν φ, e α := {X α,.} = e αµ µ e αµ = θ αµ Poisson structure frame bundle! metric averaging over internal S 2 : γ µν = g αβ e αµ e βν = g µν [e αν ] 0 = 0, [γ µν ] 0 = 4 4 gµν... SO(5) invariant!

20 H. Steinacker Emergent gravity and(in higher progress, spin, on covariant w/ M. quantum Sperling) spaces perturbed vielbein: Y a = X a + A a e a := {Y a,.} e aµ [A] µ... vielbein e αµ [A] θ αβ (δ µ β + A βρg ρµ ) + 1 r 2 θ αν θ ρσ {A νρσ, x µ } linearize & average over fiber complication: graviton is combination γ µν e αµ [A]e ν α [A] = γ µν + [δγ µν ] 0 using {P ρ, X µ } g ρµ (!) h µν := [δγ µν ] 0 = 4 ( Aµν + ρ A µρν + ρ A νρµ ) 4 basic S 4 N : ρ A µρν A µν dominates generalized S 4 Λ : modes A µν, A µρν independent, issue should be resolved (?)

21 action for spin 2 modes: expand IKKT action to second oder in A a S[Y ] = S[X]+ 2 ( g 2 TrA a ( + 1 ) 2 µ2 )δb a + 2[[X a, X b ],. ] [X a, [X b,.]] }{{} D 2 for spin 2 modes A µ θ µν A νρ P ρ +... AD 2 A h µν h µν A b coupling to matter: S[matter] M d 4 x h µν T µν auxiliary field h µν T µν! graviton doesn t propagate, due to constraint h µν A µρν h

22 exact treatment of spin 2 modes on S 4 N : Marcus Sperling & HS, arxiv: independent graviton modes h µν [A B ] c ( 1 ( + ) 2 Tµν h µν [A C ] 3c h µν [A D ] 1 3 c (1 1 2 ) T µν ) 1 T µν combined metric fluct: h µν (1 + 1 R P + 1 R 2 P 2 )T µν c = 4 g 2 vol(s 4 ) 5L 4 dim(h) NC... unphysical gravity

23 possible ways out: 1 1-loop induced gravity action h µν h µν (lin.) Einstein equations (fine-tuning...) 2 generalized fuzzy sphere S 4 Λ extra A µν modes, promising HS, arxiv: fuzzy extra dims Aschieri Grammatikopoulos HS Zoupanos hep-th/ cosmological space-times: HS, Zahn arxiv: , Sperling, HS arxiv: H 4 n : A µρν no longer dominant (?)

24 Covariant cosmological space-times & BB Lorentzian IKKT model S = Tr( [X a, X b ][X a, X b ]η aa η bb + m 2 X i X i m0x 2 0 X 0 ) choose different masses m0 2 m2 S 4 N R1,4 is solution for m 2 0 < 0 < m2 recollapsing universe η* η0 H 4 n R 1,4 is solution for m 2 < m 2 0 < 0 open universe 1.0 a(η) η* η0 0.5 fully SO(4) covariant BB from signature change & 4-form flux (H.S. arxiv: xxx)

25 summary 4D covariant quantum spaces, e.g. fuzzy S 4 N regularized higher spin theory UV cutoff, finite d.o.f. per volume Poisson structure frame bundle closely related to Vasiliev theory all ingredients for gravity, unrealistic for class. IKKT model on S 4 N gravity expected for generalized S 4 Λ cosmological H 4 n (?) (modified action) Minkowski: cosmological space-times & BB (ongoing)

26 summary 4D covariant quantum spaces, e.g. fuzzy S 4 N regularized higher spin theory UV cutoff, finite d.o.f. per volume Poisson structure frame bundle closely related to Vasiliev theory all ingredients for gravity, unrealistic for class. IKKT model on S 4 N gravity expected for generalized S 4 Λ cosmological H 4 n (?) (modified action) Minkowski: cosmological space-times & BB (ongoing) main problem: no position available...